HYDRODYNAMIC PROBLEMS ASSOCIATED WITH

SHIPS OPERATING IN RESTRICTED WATERWAYS

(2)

Effect of blockage on ship resistance

• When a ship model is tested in a towing tank of incompatible size the measured resistance is somewhat higher than that at the same speed in water of unrestricted width and depth.

• This case is the same when a ship moves in shallow and restricted waters. This increase in resistance is due to the back flow and wave retardation effect as explained previously.

• The back flow effect is usually named the blockage effect and is referred to as the corrections to the model results of towing tanks.

• There are two approaches needed to correct the boundary effect; the first is to correct the total resistance and the second to correct the speed.

Mean flow theory

The mean flow theory is used to calculate the back flow velocity (blockage correction) and it is based on the one-dimensional consideration of the conservation of energy, Bernoulli’s equation and the continuity equation.

b

a

A

hUndisturbed W.L

Depressed W.L. in way of model

Depressed WL in way of model

Section (1)Section (2)

Undisturbed WL

h

Applying the continuity equation for the free open section (1), and the section with the model (2):

Therefore,

))(( vVhbaAAV

vhbaAVhbaAVAV )()(

hbaA

hba

V

v

By neglecting terms containing

From this equation, the increase in speed can be calculated after calculating as follows:

Applying Bernoulli’s equation for the free open section (1) and the section with the model (2)

hv

aA

hba

V

v

vh

Cg

Vh

g

P

g

Vh

g

P

22

2

22

2

2

11

1

Where,

Therefore,

By neglecting the term containing

By substituting in equation

hh 1 hhh 2 VV 1 vVV 2

g

vVhh

g

P

g

Vh

g

P

2

)()(

2

22

2 2 2( ) ( )

2 2 2

V v V V v vh

g g g g

2v

g

vVh

g

vbVaVvaA

2)(

aA

hba

V

v

Therefore

By substituting in equation

)(2

g

VbaA

aVv

g

vVh

)(2

2

g

VbaA

g

Va

h

Determination of the ship resistance in shallow waterSchlichting method

Schlichting performed an analysis on the effects of shallow water on ship resistance. The analysis covered the increase in resistance in shallow water at subcritical speeds, and was for shallow water of unlimited lateral extent.

At any particular speed in deep water the wave pattern generated by the ship will have a wave length given by:

In water of depth h the same wave length would be generated at some lower speed where

and the ratio of the two speeds is

V

wL

2/2

wgLV

gVLw /2 2

wL

IV

wwI LhgLV /2tanh)2/(2

2/1)/2(tanh/ wI LhVV 2/12 )/(tanh/ VghVVI

He assumed that the wave-making

resistance in shallow water at speed

would be the same as that at speed in

deep water.

The total resistance at speed would

be found at point B by adding the

wave-making resistance to the

appropriate frictional resistance at

this speed .

IV

V

IV

WR

FhR

The other speed loss is due to the

increase in potential flow around the hull

because of the restriction of the cross-

sectional area of the waterway.

This speed loss is to be calculated in order

to find the corresponding speed in

water depth h.

Then the distance is set horizontal from point B to point C which is a point on the total resistance curve in water with a restricted depth h.

PV

hV

pIh VVV

PV

Scott method

In this method, Scott estimated the speed of a ship that would have the same resistance in deep water as it would at in water of a restricted depth h

V

hV

hwb VVVV

In order to appreciate the increase in resistance in shallow water, the following are the results of some tank testing done on the hull forms for particular Nile floating hotels:

i- Resistance data in tons for LWL = 68.0 m

B = 12.0 m

T = 1.30 m

∆ = 910 tons

7.00 8.00 9.00 10.00 11.00

Speed (Knots)

20.00

40.00

60.00

80.00

To

tal

Re

sis

tan

ce

(to

ns

)

h=3m h=5m

h=unlimited

Results of power data (BHP in KW) for a model corresponding to a ship

LWL = 67.0 m

B = 11.0 m

T = 1.50 m

∆ = 950 tons

12.00 13.00 14.00 15.00

Speed (Km/hr)

0.00

400.00

800.00

1200.00

BH

P (

KW

)

h=3.5m

h=11m

Squat of ships in shallow waters

The squat is defined as the sum of the absolute value of sinkage and half the trim

sinkage

T

½ trimsquat

In shallow water, when the static under-keel clearance is very small, and if a ship moves forward at relatively higher speeds, then grounding could occur at the bow or stern. It may be noted that grounding due to the squat of a ship could be avoided simply by reducing the speed in shallow water

The factors governing ship squat are as follows:

• The speed of the ship, which is considered to be the main factor, as the squat varies approximately with the square of the speed.

• Block coefficient, where squat varies directly with CB.

• The blockage factor, the higher the blockage factor the greater the squat.

• Water depth/ship draft ratio, the smaller the ratio the greater the squat.

Prediction of Sinkage and Trim1-Prediction based upon the slender body potential

theory

A- TuckTuck gave a solution for sinkage and trim of ships in wide shallow water

in which CS and CT are complicated expressions for the geometric characteristics of the ships under consideration.

2

2

21 nh

nh

pp

S

F

F

LCSSinkage

2

2

21 nh

nh

pp

T

F

F

LCTTrim

B- Vermeer

Vermeer developed the following simplified expressions for CS and CT in terms of the ship form coefficients:

)39452420(18

7

)98075404032(6

1

2

2 WWWPPWWPP

PWW

T

PWPWPWPW

PW

S

CiCCiCiCiCCK

C

CCiiCCCCCC

C

C- Huuska

Huuska found from experimental data for restricted shallow water, that the sinkage and trim computed with the last equations have to be multiplied by

The correction factor ε holds for

76.045.7 c

m

A

A

15.0032.0 c

m

A

A

He also developed another formula for the squat given by

Where, and are shape factors of the ship hull, and the approximate values of these factors are:

2

2

21

)5.0(

nh

nhZ

F

F

LCCSquat

ZCC

0.1

5.1

C

CZ

2-Prediction based upon the energy approach

a- Dand

Dand developed a semi-empirical method to predict sinkage and trim. This method is limited to the prediction of squat of full form ships in shallow water.

where B(x) represents the beam of the vessel on the waterline at section (x) and d(x) is the water level depression at that section.

dxxB

dxxBxdS

)(

)().(

dxxxB

dxxBxxdT

)(

)().(

B- Fuhrer and Romisch

Fuhrer and Romisch developed a method for the calculation of squat from extensive model investigation. They presented their results in the following form

This formula is applied only for

and

TL

BCS

PP

Bbcrit

2)10

(2.0

TSscrit 2.0

2/)( scritbcritcrit SSS

crit

critcrit

SV

V

V

VS ]0625.0)5.0[()(8 42

43.0/032.0 cm AA 29.2/19.1 TH

3- Prediction based upon some experimental

methods

a- Barras

Barras proposed the following formula for the calculation of the maximum squat of a ship on the basis of model and prototype measurement

For

08.232

max )( VAA

ACS

mc

mB

5.11.1 T

h

B- Soukhomel and Zass

They presented the following formulas for the calculation of sinkage :

for

for

for

296.12 Vh

TKS 4.1

T

h

296.12 KVS 4.1T

h

11.1)(0143.0 B

LK PP

0.95.3 B

LPP

C- Eryuzlu and Hausser

They derived a relationship for the maximum sinkage which was at the bow as follows:

The water depth/draft ratio varied between 1.08 and 2.78.

8.127.0

max )()(113.0gh

V

h

TBS

D- Millward

Millward carried out model experiments to derive an empirical approach to predict the squat in shallow water; and he presented the following

This formula is applied only for 0.4 < CB < 0.85 , and 1.25 < h/T < 6.0

nh

nhB

midF

FL

BC

S9.01

]46.0)(22.12[ 2

nh

nhB

bowF

FL

BC

S9.01

]55.0)(0.15[ 2

Estimation of the Limiting Speed in Shallow

Water

Yamaguchi presented a semi-empirical formula to estimate the limiting speed for tanker forms (large CB) from the requirement of under keel clearance point of view.

where is the limiting Froude number based on the ship length, for the requirement of sufficient under keel clearance.

is the draft–length ratio T/L

from test results

is the water depth-ship draft ratio h/T

is the ship beam-canal width ratio B/b

2/1

2

]

1))1(

(

)1(2[

nmeq

m

mpqFnL

nLF

p

)1/(1 eq

24.0e

m

n